The eighth term of a pair of rational recurrences

Define two sequences of rational numbers as follows: let $a_0 = 2$ and $b_0 = 3$ , and recursively define $a_n = \frac{a_{n-1}^2}{b_{n-1}}$ and $b_n=\frac{b_{n-1}^2}{a_{n-1}}$ . Find $b_8$ , leaving your answer in the exponential form $m^n/p^q$ .

Source: InternLM Lean-Workbook, record lean_workbook_33900 (Apache-2.0). The complete source proposition and its explicit variable declarations are preserved.